Answer to Question #261557 in Statistics and Probability for Shane

Question

Given a population consists of three numbers (2,5,9). 

Consider all possible samples of size 2 which can be drawn from the population.Find the standard deviation of the sampling distribution of the sample 

means.

Answer 1

Solution:

Given population = (2,5,9)

We have to assume that replacement is given here.

Then, samples of size 2 are:

(2,2), (2,5),(2,9),(5,5),(5,9),(9,9).

Mean of (2,2) = 2

Mean of (2,5) = 3.5

Mean of (2,9) = 5.5

Mean of (5,5) = 5

Mean of (5,9) = 7

Mean of (9,9) = 9

Mean of sample means $=\dfrac{2+3.5+5.5+5+7+9}{6}\approx 5.33$

Now, variance of sample means$=\dfrac{(2-5.33)^2+(3.5-5.33)^2+(5.5-5.33)^2+(5-5.33)^2+(7-5.33)^2+(9-5.33)^2}{6} \\=5.1389$

So, standard deviation$=\sigma_{\bar x}=\sqrt{5.1389}\approx2.27$